\[f_X(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left\{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right\}, \,\,\, -\infty < \mu < \infty \,\,\, \text{e} \,\,\, \sigma > 0\]
\[ \left( \displaystyle{\frac{x - \mu}{\sigma}} \right)^2 = \displaystyle{\frac{(x - \mu)^2}{\sigma^2}} = (x - \mu) \displaystyle{\frac{1}{\sigma^2}} (x - \mu) = (x - \mu) (\sigma^2)^{-1} (x - \mu),\]
que corresponde à distância quadrática entre \(x\) e \(\mu\) em unidades de \(\sigma\).
\[(\mathbf{x} - \mathbf{\mu})^t\mathbf{\Sigma}^{-1}(\mathbf{x} - \mathbf{\mu})\]
\[f(\mathbf{x}) = \frac{1}{(2\pi)^\frac{p}{2}\left|\mathbf{\Sigma} \right|^\frac{1}{2}} \exp \left\{-\frac{1}{2}(\mathbf{x} - \mathbf{\mu})^t \mathbf{\Sigma}^{-1}(\mathbf{x} - \mathbf{\mu}) \right\}\]
para \(-\infty < x_i < \infty, \,\,\, i = 1,2,\cdots, p\)
\[\mathbf{\Sigma} = \left[ \begin{array}{cc} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{array} \right]\]
\[f_{\mathbf{x}}(\mathbf{x}) = \displaystyle{\frac{1}{2 \pi \sigma_1 \sigma_2 \sqrt{1-\rho^2}}} \exp \left\{ \displaystyle{-\frac{1}{2(1-\rho^2)}} \left[ \left(\frac{x_1 - \mu_1}{\sigma_1}\right)^2 + \left(\frac{x_2 - \mu_2}{\sigma_2}\right)^2 - 2\rho \left( \frac{x_1 - \mu_1}{\sigma_1}\right) \left(\frac{x_2 - \mu_2}{\sigma_2}\right)\right] \right\}\]
Tip
À medida que \(\rho\) varia de -1 a 1:
\[(\mathbf{x} - \mathbf{\mu})^t\mathbf{\Sigma}^{-1}(\mathbf{x} - \mathbf{\mu}) \leqslant \chi_p^2(\alpha)\]
onde \(\chi_p^2(\alpha)\) é o quantil superior \(\alpha\) de uma distribuição \(\chi_p^2\) , delimita \(1 - \alpha\) de probabilidade.